Quantum Mechanics, Transition Metals, and Crystal Field Theory Study Notes

Fundamentals of Quantum Mechanics and Orbital Theory

• Quantum Mechanics Context:     * The core equation of quantum mechanics is Schrodinger's Equation, expressed as: $\hat{H}\Psi = E\Psi$     * Higher momentum ($p$) corresponds to more charge pulling toward the electron.

• Quantum Numbers:     * Principal Quantum Number ($n$): Defines the distance of electrons from the nucleus. Values are discrete integers: $n = 1, 2, 3, 4…$     * Angular Momentum Quantum Number ($l$): Determines the shape of the orbital. Range: $l = 0, \dots n-1$.     * Magnetic Quantum Number ($m_l$): Determines the orientation of the orbital in space. Range: $m_l = -l, \dots +l$.

• D Orbitals:     * Condition: $n = 3$, $l = 2$.     * Magnetic quantum numbers: $m_l = -2, -1, 0, 1, 2$.     * Characterized by having mathematically the same shape, even though they look very different visually in textbooks.     * Orientations are off-center relative to the $x$, $y$, and $z$ axes.

Transition Metals and Orbital Energy

• Transition Metals and D Orbitals:     * In a free atom or ion, all five d orbitals have the same energy level.     * Electron Configuration Example: Chromium (Cr):         * Chromium atom (Ground state): $[Ar] 4s^2 3d^4$, which rearranges to $[Ar] 4s^1 3d^5$ because these states are very close together in energy.         * Chromium ion ($Cr^{3+}$): $[Ar] 3d^3$. Electrons are removed from the $4s$ orbital first before the $3d$ orbitals.

• Complex Ions:         * Example: $[Cr(H_2O)_6]^{3+}$.         * The $Cr^{3+}$ ion acts as a Lewis acid (accepts lone pairs).         * $H_2O$ is neutral and acts as a Lewis base (donates lone pairs).         * Coordination: $Cr$ does not donate any electrons to the bond; instead, each electron pair comes from the water molecule.         * The bond is a metal-acid / water-base interaction called a coordinate covalent bond.

Crystal Field Theory (CFT)

• Mechanism of Splitting:     * Crystal Field Splitting is the separation of d orbitals into different energy levels.     * In an octahedral structure, ligands (Lewis bases) approach the metal ion along the $x$, $y$, and $z$ axes.     * Repulsion: There is repulsion between the electrons of the water (ligand) and the electrons in the d orbitals of the $Cr^{3+}$ ion.     * High Energy Level: Orbitals on the $x$, $y$, or $z$ axis experience direct repulsion, forcing them to a higher energy level.     * Low Energy Level: Orbitals shifted out of plane experience less repulsion and remain at a lower energy level. The last 3 formats of d formation are typically shifted out of plane.

• Crystal Field Splitting Energy (General):     * Symbols: $\Delta$ or $\Delta_o$ (for octahedral).     * $\Delta_o$ magnitude depends on the identity of the ligand.     * Splitting always happens in these complexes, but the value of $\Delta_o$ changes based on ligand strength.

• Spectrochemical Series (Ligand Strength):     * Weak field ligands (smaller $\Delta$): I^- < Br^- < Cl^- < F^- < OH^- < H_2O.     * Strong field ligands (larger $\Delta$): EDTA^{4-} < py \approx NH_3 < en < NO_2^- < CN^- (where $py$ is pyridine and $en$ is ethylenediamine).

Color and Light Absorption

• The Physics of Color:     * Visible light is seen when light is absorbed by the complex.     * The energy of the absorbed photon corresponds to the splitting energy: $\Delta E = E_{\text{photon}} = h\nu = \frac{hc}{\lambda}$.     * Wavelength range (in nm): Approximately $400\,nm$ (violet) to $700\,nm$ (red).

• Complementary Colors:     * The color perceived is the complement of the color absorbed.     * Example: If orange is absorbed, the complex appears blue.     * Example: If $[Co(NH_3)_6]^{3+}$ is orange, it is absorbing in the blue range due to a strong ligand/large energy gap.

• High-Spin and Low-Spin States:     * High-Spin: Caused by a weak ligand (small $\Delta$). Electrons fill all d orbitals singly before pairing ($X \sim N(\mu, \sigma^2)$). For $Co^{2+}$ ($[Ar] 3d^7$), this results in 3 unpaired electrons.     * Low-Spin: Caused by a strong ligand (large $\Delta$). Electrons pair up in the lower-energy orbitals before moving to the higher-energy ones. For $Co^{2+}$ ($[Ar] 3d^7$), this results in 1 unpaired electron.

Transition Metals in Health and Chemistry (Chapter 23)

• Section 23.1: Complex Ions:     * Metals in human health must be in usable forms, typically ionic compounds.     * Interactions between transition metals and nonmetals influence solubility as well as chemical, physical, and biological properties.     * Ligands: Molecules that form coordinate bonds with metals.

• Terminology:     * Crystal Field Splitting: Separation of d orbitals.     * Crystal Field Splitting Energy: The difference in energy between split d orbitals caused by crystal field interactions.

Concept Tests and Exercises

• Conductivity and Freezing Points:     * Question: How would the electrical conductivity and freezing points of $0.01\,m$ aqueous solutions of $[Co(NH_3)_6]Cl_3$ and $[Co(NH_3)_5Cl]Cl_2$ differ?     * Answer: The orange compound ($[Co(NH_3)_6]Cl_3$) forms 4 ions per formula unit, while the reddish-purple compound ($[Co(NH_3)_5Cl]Cl_2$) forms 3 ions. Therefore, the orange compound has greater electrical conductivity and a lower freezing point.

• Coordination Spheres:     * In $Na_3[Fe(CN)_6]$, the $CN^-$ ions occupy the inner coordination sphere of the $Fe^{3+}$ ion. $Na^+$ is the counterion. This compound has conductivity similar to $[Co(NH_3)_6]Cl_3$.     * In dissolving $NaCl$ in water, water molecules occupy the inner coordination sphere around $Na^+$ ions.

• Prussian Blue (Exercise 23.1):     * Formula: $Fe_4[Fe(CN)_6]_3$.     * Complex ion: $[Fe(CN)_6]^{4-}$.     * Oxidation state of Fe in the complex ion: $+2$ (since $6 \times CN^- = -6$ and total charge is $-4$).     * Counterion charge: $Fe^{3+}$ (Total positive charge $4 \times 3 = +12$; total negative charge $3 \times -4 = -12$).

• High-Spin vs. Low-Spin Determinations:     * $Mn^{II}$ ($Mn^{2+}$) can have high-spin or low-spin configurations.     * $Cu^{2+}$ ($d^9$ configuration) always has 1 unpaired electron regardless of spin state.     * Low-spin low-spin Mn example: $Mn(CN)_6^{4-}$ is low spin because $CN^-$ is a stronger field ligand than pyridine ($py$) found in high-spin $Mn(py)_6^{2+}$.

• Electron Counts in Octahedral Fields:     * High-spin $Fe^{2+}$: 4 unpaired electrons.     * $Cr^{2+}$: 4 unpaired electrons.     * $Co^{2+}$: 3 unpaired electrons.     * $Mn^{3+}$: 4 unpaired electrons.

• Color in Transition Metals:     * Compunds of first-row transition metals are colored because d-orbital energies split when bonding to ligands. If d-to-d transitions are possible, the compound is colored.     * $Cr(NH_3)_6^{3+}$ is yellow; $Cr(H_2O)_6^{3+}$ is violet.

Biochemistry and Medicine

• Enzymes:     * Function: Catalyze reactions by lowering the activation energy.     * Note: Not all proteins are enzymes.     * Entropy (\Delta S): For the reaction catalyzed by carboxypeptidase (hydrolysis of a peptide bond), the sign of $\Delta S$ is likely positive.

• Medical Applications of Metals:     * Toxic Metals: Coordination complexes of toxic metals can be used to kill targeted disease cells, such as cancer.     * Imaging: gadolinium-153 ($^{153}Gd$) decays by electron capture and produces gamma radiation useful for imaging.     * Cancer Drugs: Platinum- and ruthenium-containing drugs fight cancer by binding to DNA to prevent replication.